A 2D finite element analysis for contact of two cylinders.

Cananau, Sorin

1. INTRODUCTION

Contact between two elastic cylinders and spheres has important engineering applications in the behavior of many parts of machine elements, in both the macro- and the micro-scale; is more interesting to know the comportment of the bodies when sliding phenomena occurs. Thus, it is important to know the effect the contact has on the surface material and the geometry through elastic or elasto-plastic deformations and stresses field.

The cases for elastic and elastic-plastic spherical contacts have been analyzed in detail in the last three decades. Predominantly considering normal loading only, a wide array of works have analyzed the contact of rough surfaces (Liu et al., 1999).

Such analysis and results were also presented in analyzing human joints (Chen et al., 1998). Recently Nelias (Nelias et al., 2006), have explored the contact between hemispherical elastic-plastic bodies in normal loading.

However, the characteristics of normal contact as opposed to sliding contact are more complex and quite different, and this is explored in this work.

We are going to use the FEM analysis to find the behavior of elastic bodies for circular sliding. In the literature (Hamilton and Goodman, 1966) presented implicit equations and graphs of yield parameter and tensile stress distribution for circular sliding contact using the von Mises criterion for the prediction of yielding.

In this work, elastic cylinders in sliding over each other are treated as whole bodies. The classical model of "elliptical contact against a rigid flat" is the consequence of the elastic Hertzian theory as is shown in Johnson and Moon (Johnson, Moon, 2006)

But for an analysis which includes sliding contact conditions this theory has no physical grounds or mathematical proof once friction phenomena takes place, certainly not when the two sliding bodies have distinct material properties. These parameters are particularly critical in understanding the sliding phenomenon.

By means of FEA actual sliding is simulated, taken into account the different position (eccentricity) of the cylinders, for wherein the two interfering bodies are both fully modeled, without resorting to the common model of an equivalent body against a flat. This is important when sliding takes place between different materials.

2. 2-D FINITE ELEMENT MODEL

2.1 Hypothesis

Following are the assumptions that are used to simplify the problem:

(1) The contact model of Hertz is taken into account.

(2) The two cylinders are considered to be infinitely long in the direction perpendicular to sliding, which enables the FE model to be in 2D under the assumption of plane strain behavior.

(3) The sliding bodies are idealized to have elastic-perfectly plastic behavior.

(4) It is assumed that the mesh validated up to the onset of plasticity is tested for further analysis of the elastic-plastic regime.

(5) Sliding is simulated as a quasi-static process, i.e. time-dependent phenomena are not analyzed. The quasi-static process is a step by step process.

In the elastic domain and up to the limit of plasticity, the Hertzian solution (Nosonovsky, Adams, 2000) provides critical values of load, contact half-width, and strain energy. The hardness is not an unique material property as it varies with the deformation itself as well as with other material properties such as yield strength or the elastic modulus.

In this paper the critical vertical interference, [[omega].sub.c], as derived by Green (Green, 2005) for cylindrical contact, is taken into account. This parameter is calculated by comparing the stress value (maximal energy criterion) with the yield strength, Sy. The relation for interference is given as:

[[omega].sub.c] = r[(C x [S.sub.y]/E).sup.2][2ln(2E'/C x [S.sub.y]-1)] (1)

Where

C--critical yield stress coefficient

E--modulus of elasticity

E'--equivalent modulus of elasticity

r--radius of cylinders

[S.sub.y]--yield strength

The values for critical yield stress coefficient C is found using the Poisson's ration material property, v.

C = 1/[square root of 1 + 4(v-1)v], v [less than or equal to] 0.1938 (2)

or

C = 1.164 + 2.975v - 2.906[v.sub.2], v > 0.1938 (3)

For different materials, the value of coefficient C is given as [CS.sub.y] = min(C([v.sub.1])[Sy.sub.1],C([v.sub.2])[Sy.sub.2]).

2.2 Material properties, geometry and calculus

In this work, the values for the terms in equations (1)-(3) are calculated for a steel material with properties shown in the table Tab. 1. The materials properties are equivalent to a common steel.

Since all the values are subsequently being normalized by the aforementioned Eqs. (1)-(3), the further results apply for a geometry scale (with the hypothesis of homogeneous and isotropic continuum mechanics) is not important at a macroscale view; therefore, the radii for the cylinders in the FE model are subjectively chosen. The Finite Element model is designed for contact of two semi-circles with quasi static sliding relative motion, modeled to transverse one over the other with a preset vertical interference [omega]. The model is shown in Fig. 1. The commercial FEA software DS COSMOSWORKS (SOLIDWORKS 2007) is used to perform the analyses. The mesh is constructed using four node triangular elements, surface-to-surface contact elements and specific sliding conditions.Various mesh schemes are tried to achieve convergence. The optimized model has 103206 nodes, 46570 elements, and 320 contact elements in the region of interest. The mesh was validated first for classical solution of a purely aligned normal elastic contact (non-sliding), with Sy=0.78 GPa, and results are compared with the analytical solution obtained by Green.

[FIGURE 1 OMITTED]

3. ANALYSIS AND RESULTS

For the case were both cylinders are modeled with the same material properties and the same geometry, as shown in Tab. 1, the deformation pattern followed by the two is identical. The position of the maximum vertical displacement on the surface of the cylinders moves along as sliding progresses because of the curvature of the two cylinders as is shown in Fig. 2.

The result is obtained for a range of preset normalized vertical interferences, [[omega].sub.N]=1.5, where [[omega].sub.N] is the non-dimensional vertical interference between cylinders, [omega]/[[omega].sub.c]. For the next analysis we perform many tests for diffrenet values of [[omega].sub.N]. In the Fig. 3 is shown the result for the interference [[omega].sub.N]= 15, (ten times greater).

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

For the final step we show the result for the case of sever sliding conditions where the plastic deformations occurs and where the influence of the tangential forces is present in a subsurface.

[FIGURE 4 OMITTED]

4. CONCLUSION

For high values of vertical interferences the large magnitudes of stresses are found in the contact of the cylinders for both frictionless as well as frictional sliding. So, it is thus reasonable to conclude that for high vertical interference values, these regions with such accumulation of stresses will be the cause of failure. [paragraph]This analyze is an important tool to know the effect the contact has on the surface material and the geometry through elastic or elasto-plastic deformations and stresses field.

5. REFERENCES

Chen, J. et al.. (1998). Stress analysis of the human temporo-mandibular joint, Med Eng. Phys. 20 (8) (1998) 565-569

Green, I.(2005) Poisson ratio effects and critical values in spherical and cylindrical Hertzian contacts, Int. J. Appl. Mech. 10 (3) (2005) 451-462

Hamilton, G.M.; Goodman, L.E. (1966). The stress field created by a circular sliding contact. J. Appl. Mech. 33 (1966), 371-376

Johnson, J.A.; Moon, F.C. (2006). Elastic waves and solid armature contact pressure in electromagnetic launchers. IEEE Trans. Magn. 42 (3) (2006), 422-429

Nelias, D.; Boucly, V.;Brunet, M. (2006). Elastic-plastic contact between rough surfaces: proposal for a wear or running-in model. J. Tribol. 128 (2) (2006), 236-244

Nosonovsky, M.; Adams, G.G.. (2006). Steady-state frictional sliding of two elastic bodies with a wavy contact interface. J. Tribol. Trans. ASME. 122 (3) (2000), 490-499

Tab. 1. Materials properties and geometry for cylinders in
contact

Material M, Elasticity Poisson's Radius, r
([M.sub.1], modulus, ratio, v ([r.sub.1],
[M.sub.2]) E, ([E.sub.1], ([v.sub.1], [r.sub.2])
 [E.sub.2]) [v.sub.2])

AISI 1045 205 GPa 0,28 40 mm
Steel